Algorithmic Issues in Modeling Motion

Pankaj K. Agarwal, Leonidas J. Guibas,Herbert Edelsbrunner, Jeff Erickson, Michael Isard, Sariel Har-Peled,

John Hershberger, Christian Jensen, Lydia Kavraki, Patrice Koehl, Ming Lin,Dinesh Manocha, Dimitris Metaxas, Brian Mirtich,

David Mount, S. Muthukrishnan, Dinesh Pai, Elisha Sacks,Jack Snoeyink, Subhash Suri, Ouri Wolfson ∗

Abstract

This report presents the results of the workshop on Algorithmic Issues in ModelingMotion, funded by NSF and ARO, held on August 6 and 7, 2000 at Duke University,Durham, NC. The report identifies research areas in which motion plays a pivotal role,summarizes the challenges that lie ahead in dealing with motion, and makes a numberof specific recommendations to address some of the challenges presented.

∗Pankaj K. Agarwal, Duke University, pankaj@cs.duke.edu; Leonidas J. Guibas, Stanford University,guibas@cs.stanford.edu; Herbert Edelsbrunner, Duke University, edels@cs.duke.edu; Jeff Erickson, University ofIllinois, Urbana-Champaign, jeffe@cs.uiuc.edu; Michael Isard, Compaq Research Labs, misard@robots.ox.ac.uk;Sariel Har-Peled, University of Illinois, Urbana-Champaign, sariel@cs.uiuc.edu; John Hershberger, Mentor Graphicsjohn_hershberger@mentorg.com; Christian Jensen, University of Aalborg csj@borg.cs.auc.dk; Lydia Kavraki, RiceUniversity, kavraki@cs.rice.edu; Ming Lin, University of North Carolina, Chapel Hill, ming@cs.unc.edu; DineshManocha, University of North Carolina, Chapel Hill, dm@cs.unc.edu; Dimitris Metaxas, University of Pennsylva-nia dnm@central.cis.upenn.edu; Brian Mirtich, MERL mirtich@merl.com; David Mount, University of Maryland,mount@cs.umd.edu; S. Muthukrishnan, AT&T, muthu@research.att.com; Dinesh Pai, University of British Columbia,pai@cs.ubc.ca; Elisha Sacks, Purdue University, eps@cs.purdue.edu; Jack Snoeyink, University of North Carolina, ChapelHill snoeyink@cs.unc.edu; Subhash Suri, University of California, Santa Barnara, suri@cs.ucsb.edu; Ouri Wolfson, Uni-versity of Chicago wolfson@eecs.uic.edu.

1 Introduction and Background

Motion is ubiquitous in the physical world. Computational models of physical objects andprocesses, from protein folding in biology, to assembly planning in manufacturing, from connectivityin mobile communications networks to location based services in spatial databases, from ecologicalmodels in environment science to galactic evolution in astrophysics, must deal with representingmobile data and their evolution over time. It is instructive to draw a comparison with anothermodality that permeates modeling the physical world as much as motion, namely shape. Over theyears computational shape representations have been intensely studied and a variety of approacheswere developed for modeling the different shapes that arise in applications. Entire disciplines,such as CAGD (computer-aided geometric design) evolved, in this case devoted to the of study ofthe smooth shapes needed by the automotive and aerospace industries. What is striking when wecompare motion to shape, however, is how rich and complex the space of representations for motioncan be. For one, motion is often described intensionally — by specifying what the motion needs toaccomplish, and not what the motion is. For another, motion happens over time and in the physicalworld unpredictable events can happen that change the evolution of a system — this on-line characterof motion must be modeled as well.

Both military and civilian applications (digitized battlefields, automatic target recognition, mo-bile communications, virtual environments, animation, physics-based simulation, animation of de-formable objects — to name a few) have raised a wide range of algorithmic issues in modelingmotion and coping with continuously moving objects. Proper algorithmic techniques are lacking fordealing with many of these issues. Despite a flurry of activity in several areas in the last few yearson modeling motion and on developing efficient algorithms that deal with moving objects, a unifiedalgorithmic theory of motion that permeates across multiple disciplines and applications has beenmissing. A workshop, sponsored by NSF and ARO, held on August 6 and 7, 2000 at Duke Universitywas aimed at bringing together a set of researchers to survey the state of the art in motion modelingand to propose initiatives that would create a firm algorithmic theory for the processes of acquiring,modeling, reasoning about, planning, manipulating, and executing motion.

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2 Goals

Geometric computing lies at the core of many areas highlighted in the IT2 initiative: informationvisualization, advanced science and engineering computation, and computational and algorithmicmethods. Efficient algorithms and data structures for handling geometric data are needed in almostevery computational field that deals with the physical world, including computer graphics, computervision, robotics, geographic information systems, spatial databases, molecular biology, and scientificcomputing — not to mention entertainment and computer games.

Although a vast amount of work has been done on modeling, analyzing, searching, processing,and visualizing geometric objects, most of the work has by and large focussed on handling staticobjects and on dynamic data structures, which can update the information as objects are insertedor deleted at discrete times. These approaches are, however, not suitable for handling moving ordeforming objects because either the algorithm has to be executed continuously, or the solutionreturned will be at times obsolete. The on-line nature of motion makes it difficult to consider timesimply as another dimension and regard the moving object as a static object in space-time. Algorithmsthat can handle continuous motion, what we call kinetic algorithms, are needed in such applications.

Most of the work to date on modeling motion and kinetic algorithms has been scattered among anumber of fields. This report argues that a synergy among these areas is needed, since many issuesare common to all the areas; furthermore many real-world applications require the integration oftechniques from a wide range of disciplines. This report includes sections devoted specifically tocomputational geometry, mesh generation, physical simulation, biology, computer graphics, com-puter vision, robotics, spatio-temporal data-bases, and mobile wireless networks. It concludes byhighlighting a number of challenges common across several of these fields and making a set ofrecommendations to further advance the integration across disciplines and push the state of the artin motion algorithms.

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3 Areas and Problems

We have identified nine main areas in which motion plays a pivotal role. Although there areseveral common issues that arise in many of these areas (for instance, motion representation, dealingwith uncertainty, etc.), some of the issues are particular to specific domains. This section presentsthe state-of-the-art concerning motion modeling in each of these areas and describes some of themain open problems in each of them.

3.1 Computational Geometry

Earlier work on moving objects in computational geometry, initiated by a paper byAtallah in 1985 [7],focused on bounding the number of combinatorial changes in geometric structures as the objectsmove along prescribed trajectories [6]. In the late 1980s algorithms were developed for computingthe changes in geometric structures as objects undergo motion. However, these results assumean off-line model of motion in which the trajectories are given in advance, which is a relativelyuncommon situation. In practice, researchers were using fixed-rate sampling methods to handlemoving objects, but without a principled to way to select the proper time step. The kinetic datastructure (KDS) framework introduced by Guibas et al. [9] alleviated many of the problems withthe off-line and fixed-rate-sampling methods. The main idea in the kinetic framework is that eventhough objects move continuously, the relevant combinatorial structure changes only at certaindiscrete times and therefore one does not have to update the data structure continuously. The validityof the combinatorial structure is certified by a number of elementary assertions about the state ofthe system called certificates. Kinetic updates are performed on the data structure only when certainkinetic events occur, corresponding to certificate failures. In contrast to fixed-time-step methods, inwhich the structure is updated at fixed time intervals and therefore the fastest moving object gatesthe time step for the entire system, a kinetic data structure is based on events determined by localizedconditions that can be monitored at different rates.

Kinetic data structures have led to efficient algorithms and data structures for a wide range ofproblems. However, in order for KDS and other motion models to be made more realistic, severalissues need to be addressed.

Motion sensitivity. The motions of objects in the world are often highly correlated and it behoovesus to find representations and data structures that exploit such motion coherence. It is also importantto find mathematical measures that capture how coherent motions are and then use this coherence asa parameter to quantify the performance of motion algorithms. If we do not do this, our algorithmdesign may be aimed at unrealistic worst-case behavior, without capturing solutions that exploit thespecial structure of the motion data that actually arise in practice. Most kinetic algorithms to datehave been analyzed under the model where each object follows an independent smooth trajectorydefined by a small set of parameters, and the algorithm’s efficiency is measured by the worst-casebehavior over all such possible motions. A challenging issue is to develop a class of kinetic motion-

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sensitive algorithms, whose performance can be expressed as a function of how coherent the motionsof the underlying objects are.

Trade-offs between complexity measures. The performance of a kinetic algorithm can be mea-sured by the number of events it processes, the time spent in updating the structure at each event,and the size of the structure. Although optimal or near-optimal algorithms under these criteria havebeen proposed for a number of kinetic problems, including convex hull of a point set in the planeand closest pair, no optimal algorithms are known for a wide range of other problems e.g., convexhulls in R

3, triangulation of points in R2, range searching, etc. In such cases, one would like to

obtain a trade-off between different complexity measures. A natural trade-off would be between thecompactness and the efficiency of a KDS. This is similar to the trade-off between the size and thequery time of a static data structure — a query can be answered more efficiently by allowing morespace. A similar issue is to improve the efficiency of a kinetic algorithm by maintaining a geometricstructure implicitly. For example, we know of techniques for maintaining the triangulation of a pointset in the plane implicitly by structures that process only quadratic number of events, but no suchbounds are known if is required to maintain a triangulation explicitly.

Another possible trade-off is between efficiency and accuracy. How much efficiency can begained by maintaining a geometric structure approximately? For example, it is known that thediameter of a set S of points (the maximum distance between a pair of points in S) moving in theplane can change quadratically many times [3], but recently it was shown [5] that if we maintainthe diameter approximately, the number of events only depends on the approximation factor (and isindependent of the number of points). Although some preliminary work has been done on tradingoff efficiency with accuracy, a more general theory is needed for approximating motion.

Flexible scheduling. The current KDS implementations maintain a global priority queue that storesthe future events at which the structure may need to be updated. The events are processed in thesorted temporal order. The correctness of these methods relies on the assumption that all the eventshave been processed in this order. Under algebraic motion an event is typically a root of a polynomialof fixed degree, and thus we have to compute the roots of a polynomial. If we use floating-pointarithmetic or numerical methods to estimate the roots, we compute them only approximately andcannot ensure that the events have been processed in the sorted order. On the other hand, using exactarithmetic and algebraic methods (such as Sturm sequences) for computing or isolating roots is quiteexpensive. Another source of difficulty in processing events in the sorted order is degeneracies inthe input — a degeneracy causes multiple events to occur simultaneously. Little attention has beenpaid so far to these problems.

In many physical simulations the motion law of the objects is specified by an ordinary or partialdifferential equation. In such contexts polynomial trajectories can be at best short-term approxi-mations of the actual trajectories, useful for conservative estimates of certificate failure times. Thecorrectness of the approximation needs to be certified and tracked as well.

It is desirable to develop a framework where event-based scheduling can be intermixed withfixed time-stepping. The latter can be used whenever explicit motion prediction, apart from fullintegration of the equations of motion, is difficult. The current KDS repair mechanism strongly

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depends on the assumption that it is invoked to repair a single certificate failure. In a time-step basedscenario multiple certificates may fail; we need techniques that can repair a structure after multiplecertificate failures.

Canonical vs. non-canonical structures. The complexity measures mentioned earlier are moresuitable for maintaining canonical geometric structures, those that are uniquely defined by theposition of the data, e.g., convex hull, closest pair, and Delaunay triangulation. In these cases thenotion of external events (those that change the combinatorial structure of the attribute of interest) iswell defined and is independent of the algorithm used to maintain the structure. On the other hand,suppose we want to maintain a triangulation of a moving point set. Since the triangulation of a pointset is not unique, the external events depend on the triangulation being maintained, and thus depend onthe algorithm. This makes it hard to analyze the efficiency of a kinetic triangulation algorithm. Mostof the current approaches for maintaining non-canonical structures artificially impose canonicalityand maintain the resulting canonical structure. But this typically increases the number of events.Another drawback of imposing a canonicality condition is that in it might make the structure global,which in turn would make the task of designing a distributed algorithm considerably harder.

In general, current techniques are too weak to analyze the performance of non-canonical struc-tures and better techniques are needed. In [1] we made some progress in this direction, but thesetechniques are very problem specific. A better model for analyzing the lower bounds is needed.Some of the analysis techniques developed for analyzing the on-line algorithms could prove useful.

Decentralization. When we want to simulate motion over a large scale, e.g., the flow patternsof cars in a city or a chemical reaction among several macromolecules, the motion computationmay have to be distributed over several processors/machines. Furthermore, if we are confirmingcertificates from sensor data, the sensors themselves may be geographically distributed over a largearea and may be also constrained by communication bandwidth and power requirements. In suchsettings the normal centralized way of implementing KDSs cannot be used, as the communication,processing, and power capabilities of the central node will severely limit the scalability of the system.Most of the existing approaches assume the existence of a central event queue that keeps track of allthe changes. In a distributed environment, reasoning about the state of the world and tracking theattribute of interest must take place in a distributed manner. Each processing node will be responsiblefor only some of the mobile data and will communicate information with other nodes as necessary.How to develop such algorithms even for simple problems remains a challenging issue, despite afew recent attempts in this direction.

The number of combinatorial changes. Although optimal or near-optimal bounds are known onthe number of changes in the basic geometric structures such as convex hull, closest pair, diameter,and width of a point set, such bounds have remained elusive on many other structures, includingDelaunay triangulation, minimum spanning tree, and alpha shapes. Moreover, most of the workhas focused on bounding the number of changes in the worst-case, which rarely occurs in practice.There has been some work on bounding the number of changes when the trajectory of each point israndomly chosen from a reasonable distribution. It would be useful to study the number of changes

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under more realistic assumptions on motion, e.g., when the motion of the points is correlated, or todevelop a model in which the bounds are proved as a function of the complexity of the motion ofobjects.

3.2 Mesh Generation

Many physical and engineering problems are modeled by defining a set of partial differential equa-tions over a continuous domain. Since it is infeasible to perform computer simulations over acontinuous space, the domain is discretized by decomposing its interior into a mesh of simple andwell-shaped elements such as boxes or simplices, the differential equations are approximated overthe discretized domain using finite difference, element, or volume techniques, and then solved nu-merically. The error of a solution is estimated, and if necessary, the mesh is refined and the numericalsimulation repeated. Not all meshes are equally good, because the error of discretization dependson the geometric shape and size of the elements of the mesh and the computational complexity ofthe simulation depends on the number of elements in the mesh and their overall quality. This has ledto vast literature on mesh generation; see [10] for a recent survey.

In many applications, ranging from weather modeling to seismic analysis, from protein foldingto heart modeling, the domain evolves with time, which raises the problem of mesh generation overtime-varying domains. Depending on the application, either the mesh is updated dynamically overtime, or time is considered as another dimension and the mesh is generated over the space-timedomain.

Dynamic meshes. To maintain the mesh we must trace changes of the shape and repair the meshat places where it becomes inadequate. We see at least three types of changes:

(i) Changes of shape, which can usually be accommodated by moving mesh vertices (and thusimplicitly cells).

(ii) Changes of curvature, which cause distortions of the homeomorphism between the mesh andthe underlying domain. In the case of triangulations we can repair such distortions by locallyinserting or deleting simplices.

(iii) changes of topology, which require a local reorganization of the mesh connectivity.

In all cases, we require good mechanisms to predict or detect when these changes happen, as wealready discussed for kinetic certificates in Section 3.1

Algorithms that maintain meshes under all three types of changes are rare [14], and they are sen-sitive to how quickly changes can happen. This gives rise to a hierarchy of motions or deformations,ordered by difficulty in maintaining a mesh algorithmically. A deformation may be best understoodby considering the trajectory of the domain through a higher-dimensional space-time. For example,a deforming 2-dimensional surface in R

3 sweeps out a 3-dimensional manifold in R4. The speed of

the deformation is then captured by the slope of the manifold in the direction of time.

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Meshing space-time domains. A possible alternative to dynamic meshing is meshing directlyin space-time, which trades the dynamic nature of the problem for an extra dimension. Space-time meshing algorithms are becoming more desirable now as numerical algorithms for space-timeproblems are being developed. Recently, space-time discontinuous Galerkin methods have beenproposed [30]. These methods rely on the domain of influence for dynamic data. The domainof influence for a point p in the space-time domain is a cone with apex p whose boundaries aredetermined by the maximum speed; the cone specifies the region in which the point p can lie in thefuture. In these methods, one needs to develop meshes that satisfy the so-called cone constraint,which requires the dihedral angle of each interior mesh face with respect to the space domain to bebounded by a specified parameter, which depends on the boundary of the cone. However, unlikemany other techniques, the discontinuous Galerkin methods impose no restriction on the shape ofelements of the mesh. Despite some recent work on one- and two-dimensional spatial domains, theproblem of meshing over space-time domain remains largely open. Here are a few open problems:

• Many of the existing methods divide the time into fixed length intervals and construct the meshwithin each layer. Can a more adaptive approach be developed?

• The current methods generate too many elements for the mesh; no technique with provablebound on the size of the mesh is known.

• In many material flow problems, the cone constraint is skewed or anisotropic. The currenttechniques assume uniform cone constraints. This brings the issue of modeling increasinglycomplex motion hierarchically, as in Section 3.1.

• Sometimes, e.g., in presence of shock waves, the cone constraint is discontinuous or it changeswith time. In such cases, the mesh has to be adapted with time.

3.3 Physical Simulation

The applications of physical simulation can be divided into two broad categories. In one category areapplications requiring physical simulation to reason about the real world for engineering, verification,and prediction. Here, accuracy is key, and many of the challenges involve the development ofaccurate models. Many of these, such as finite-element or boundary-element models, require theuse of meshing techniques as discussed in Section 3.2. The second category of applications requirephysical simulation for producing ‘plausible’ motion; these applications include animation, games,content creation, and some kinds of education and training. Here, since the constraints on accuracy aremore lax, modeling is less of a problem, and efforts have focused on other issues such as unstructuredinteraction and real-time speeds. Below are four challenges related to motion in physical simulation;the first three are relevant to both categories of applications, and the final one is relevant to thenon-engineering applications.

One grand challenge of physical simulation is the combination of different state representations tosolve complex, heterogeneous problems. Consider the different ways motion is represented throughvelocities. The velocities of all points on a rigid body are simultaneously described by the linearvelocity of a single point on the body and an angular velocity. On the other hand, the velocity of a

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deformable body is often represented by the linear velocity of a number of node points attached to thebody. Finally, the velocity of fluid might be represented by linear velocities at lattice points fixed inspace. How are these representations to be harmonized when studying, for example, the turbulenceinduced by a rigid body moving through a fluid, or the wrinkling of the clothes on an articulatedhuman figure? Some progress has been made, but many problems remain in modeling interactionsand constraints between objects with different underlying state and motion representations.

A second challenge is to design simulation systems than can effectively choose appropriate mo-tion models and algorithms for the given situation. Perhaps a rigid body model is a reasonableapproximation, or maybe a linear or even non-linear deformable model is needed. Inertial effectsmay or may not be negligible. There are a variety of ways to model friction with different prop-erties and computational costs. Even after the model has been chosen, there might be alternativealgorithms. For example, there are at least three families of methods for modeling rigid body con-tact (analytic/LCP formulations, penalty methods, and impulse-based methods), each with differentstrengths and weaknesses. By and large, today’s simulation systems use a single model and a singlealgorithm for a given class of dynamic objects. Better if they could choose more judiciously amongthem on the fly. To do this, a system must know what are the important results of the simulation, whatare the allowable tolerances, what are the sources of error in each algorithm, what are the bounds onthese errors, and how do the errors propagate over time.

A third challenge is to parallelize simulation algorithms so that they can be scaled to biggerproblems by adding more computational units. This is already possible for certain types of simu-lations involving finite element methods and computational fluid dynamics. It is harder for moreunstructured systems such as those involving rigid bodies. Often, such systems can not be staticallypartitioned into independent subsystems, and furthermore the parts of the system that affect anotherpart vary over time. A partitioning among CPUs needs to be a dynamic one. Better bounds on themotion of objects over time could make maintaining this partition easier. Also to be considered areload balancing issues: equally sized partitions are most useful.

A final challenge in physical simulation is to give better control to the user. Particularly whenused for non-engineering applications, simulation is praised for producing complex motion whilecursed for being an unwieldy, unpredictable tool. Despite living in a physical world, humans arepoor at designing simulations to give them ‘what they want;’ in many cases the desired outcomes arenot even physically possible. Despite a lot of work in this area, users still need better handles andhooks into the physical simulation process in order to steer its course. Often the user’s desires areexpressible in terms of motion constraints. Representations for these constraints are needed that areboth intuitive to humans and which lead to tractable computer solutions. Also needed are algorithmsthat can let go of the true physics just enough to meet the constraints.

3.4 Biology

Molecules move and deform. Chemical processes essential to life critically depend on the abilityof biomolecules to adopt different shapes over time. If we could realistically describe molecularmotions, we would greatly improve our understanding of these temporal processes, including pro-tein folding and molecular interactions (such as the assembly of protein complexes, protein-ligand

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docking, etc.), representing some of the most important unsolved problems in biology. Simulatingthe motions of molecules poses its own challenges, beyond those covered in earlier sections.

Molecular dynamics simulations. At the atomic level, the physical principles underlying molec-ular motions are fairly well understood: ligands and proteins have primarily torsional degrees offreedom, and the forces on an individual atom can be expressed as sums of well-known potentials.Yet molecular dynamics simulations of a molecular system are still far too expensive computationallyto allow us to follow a process over its complete time period. For example, a protein folds withina few seconds, while the longest published molecular dynamics simulation of a protein covers onemicrosecond. The reason that accurate molecular dynamics simulations remain a major challengein computational biology is that they involve thousands of degrees of freedom in the molecule ofinterest, require many evaluations of potential energy functions that can be expensive to compute,and in addition need to take in account the solvent environment. In order to make progress, wemust develop novel ways to represent deforming shapes and track their evolution over time, findtechniques for describing the physics in terms of higher-level units than individual atoms, and useefficient approximations whenever justified to do so. More specifically, we need:

Implicit solvent potential energy functions. Molecular dynamics simulations that include a largenumber of water molecules around the solute remain the state of the art in this field. They are,however, inefficient, since a large fraction of the computing time is spent calculating a detailedtrajectory of the solvent molecules, even though it is primarily the solute behavior that is ofinterest. Several semi-analytical implicit treatments of solvent have been proposed. They allrely on an energy term that is related to the solvent exposed surface area of each atom of thesolute. Inclusion of such terms in a molecular dynamics simulation requires the calculation ofaccurate surface areas, as well as their analytical derivatives with respect to atomic position.Fast analytical methods are needed for these calculations [12].

Efficient updating procedures. Molecular dynamics simulations are based on solving the Newtonequations of motion for all atoms of the system considered. These equations are solvednumerically, using very short time steps of the order of a fraction of a femtosecond. Each steprequires the evaluation of the total energy function of the system, as well as its derivatives withrespect to atomic position. For standard energy terms such as Lennard-Jones and electrostaticsinteractions, only a fraction of all pairs of atoms need be considered. A crucial aspect for fastmolecular dynamics simulation is to have access to fast algorithms for building and updatingthese interacting pairs of atoms.

Hierarchical representation of proteins. Very long simulation of molecular dynamics require ap-proximations. One approach is to simplify the representation of the molecule of interest:simplified models of proteins have proved both popular and effective until the present time. Asignificant weakness of such models however is their inability to deal with topological featuresof the molecule in a meaningful way. We need new methods that build simplified models withgeometrical persistence.

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A motion-planning approach to protein folding. Molecular dynamics simulations follow molec-ular motions by solving a deterministic or stochastic system of equations. They provide very detailedanalysis of short and usually small amplitude motions, with reasonable agreement with experimentaldata coming from structural biology. While it is expected that improvements of the computer tech-nology and in the algorithms applied for solving these equations will increase the time span coveredby these simulations, time scales on the order of milliseconds and seconds are still out of reach. Aswe already remarked, unfortunately these are the time scales of most interest for many importantbiological processes. Different approaches specific to these problems need to be explored.

There are large and ongoing research efforts whose goal is to determine the native folds ofproteins (the protein structure prediction problem). Most approaches proceed in two steps. Firstly,a large collection of possible conformations (or decoys) for the protein of interest is built. Secondly,these conformations are screened using potential energy functions and the ‘best’ models definethe predicted structures for the protein. In these procedures, the focus is set on thermodynamics(definition of low energy states) and not on the kinetics of the protein folding process (i.e., howthe protein folds to its native conformation). A better understanding of the latter should not onlyfacilitate protein structure prediction but also provide insights on how proteins function. Motionplanning and more specifically probabilistic roadmap methods provide one approach to solving thisproblem. Application of such techniques to the protein folding problem requires:

Falsifiability studies. Motion planning usually starts by sampling the moving object’s configurationspace (C-space), and retaining those conformations that satisfy certain feasibility requirements.Roadmap nodes are then connected by finding the path of minimum energy between the startingpoint and the goal. Application of such a strategy to the protein folding problem is not intuitiveand should be extensively tested. Interestingly, recent experimental results suggest that proteinfolding mechanisms and landscapes are largely determined by the topology of the native stateand are relatively insensitive to details of the interatomic interactions. This dependence on lowresolution structural features suggest that it should be possible to describe the physics of thefolding process using relatively low resolution models. It remains that initial studies shouldfocus on proteins for which kinetic data are available, as well as on proteins whose dynamicshave been described.

Algorithmic developments. In the case of protein folding, the C-space is large, since the proteinhas a very large number of degrees of freedom. Fast algorithms for extensive sampling ofthe C-space of proteins must be developed. The protein folds based on geometric constraints(i.e., no self-collision) but also energetic constraints. Realistic energy functions are thereforeneeded, such as those used for molecular dynamics simulations — a point already mentioned.Algorithms are also needed to select roadmap nodes from possible configurations of the proteinin its C-space. Such selection can be based on energetic and/or geometric criteria.

3.5 Computer Graphics

Computer graphics is concerned with object geometry and appearance models, scene description,and image generation. As the field of computer graphics and virtual environments has evolved, there

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has been increasingly interest in acquiring and tracking the motion of users, as well as in simulatingdynamic and changing environments. We categorize the research challenges concerning motion incomputer graphics into two major areas: interactive display and motion generation.

Interactive display. In traditional polygon rendering of massive geometry datasets, several render-ing acceleration techniques have been proposed, including visibility culling, model simplification,and image-based techniques. Over the last few years, model simplification for static environmentshas been extensively studied; see [20, 26] and references therein. However, little has been donefor geometric simplification of complex, dynamic environments. Erickson and Manocha [17] haverecently proposed an automatic technique based on ‘‘hierarchical levels of detail’’ (HLOD) to au-tomatically simplify large dynamic environments. It works well for datasets with modest amountof movement. However, further study is required for view dependent simplification of dynamicenvironments consisting of relatively large movement. Similarly, little research has been done onvisibility or occlusion culling in dynamic environments. Although kinetic data structures (as dis-cussed in Section 3.1) for spatial decompositions such as binary space partition trees have beenproposed [4, 2] and implemented, several robustness and efficiency issues must be addressed beforethey become suitable for general use.

A difficulty in adapting the known simplification and occlusion-culling techniques to handledynamic environments is that they require sophisticated preprocessing steps, which makes the updatesquite expensive. Simplification and occlusion-culling techniques that involve little preprocessingare needed for dynamic environments. Furthermore, often a data structure is specially designed foreach rendering acceleration technique. It is impractical to integrate all different types of renderingacceleration techniques using multiple data structure of real-time rendering of complex geometricdatasets. Thus it remains a challenge to design a unified data structure for combining multiplerendering acceleration techniques (e.g. visibility culling, geometry simplification and image-basedrendering) for interactive display of a massive environment, using only one type of data structure toachieve near-optimal performance.

Besides polygon-based rendering, volume rendering and image based rendering have recentlyreceived much attention. How to extend these techniques to dynamic environments remains a chal-lenging problem.

Extending the frontier of man-machine interaction, haptic rendering augments graphical andaural display by enhancing the level of immersion and accommodating the sense of touch. In orderto create a sense of touch between the user’s hand and a virtual object, contact or restoring forcesare generated to prevent penetration into the virtual model. This is computed by first detectingif a collision or penetration has occurred, then determining the (projected) contact point on themodel surface. If penetration has occurred, then it is often necessary to compute the penetrationdepth for computing the restoring forces. Some of open research issues are (1) design of proximityquery algorithms capable of running at the KHz rates required by commercial haptic devices, (2)haptic display of textured surfaces — i.e. fast and accurate simulation of friction; (3) simulation ofdeformation between flexible bodies for enabling real-time haptic interaction.

In order to immerse a user in a virtual environment (VE), motion tracking and prediction isessential for man-in-the-loop interaction with VEs. Tracking user motion, including head, hand and

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body movement, for real-time mobile interaction with virtual environments is of great importance.Augmented reality (superimposing a virtual scene over a real, physical environment), registering mo-tion of synthetic environment with the physical world undergoing motion presents another challengeto researchers in motion modeling.

Motion generation. The following four mechanisms are commonly used to generate and synthesizemotion for computer graphics and virtual environments: animation by keyframing and kinematics,playback using motion capture, physically-based simulation, and high-level behavior control.

A scalable and automatic technique for motion editing is one of the active research areas foranimation using keyframing. Other problems include specifying and representing deformation forkinematics-based approach to synthesize motion. Despite many years of investigation in this area,developing accurate and robust methods for stable motion tracking and registration based on captureddata remains an open research challenge. Once the captured data is available, fitting the recordedmotion with any animated character also presents several interesting problems.

For physically-based simulation, there are several outstanding research issues. Some of theminclude dynamic simulation and control of complex systems (articulated, flexible bodies), modelingof heterogeneous and deformable materials, use of simulation level of details (as an analog togeometric levels of detail), and the design of proximity query algorithms based on multiresolutionrepresentations. Several of these issues have already been discussed in Section 3.3. Some of researchproblems for high-level behavior control include task and motion planning of autonomous agents,control of articulated bodies, and high-level behavior learning based on captured motion libraries.

3.6 Computer Vision

Shape and motion. Vision processes data representing motion in the physical world. Sensor noise,calibration, and uncertainty are novel issues that have to be addressed. Motion analysis in computervision can very crudely be divided into scene-based and object-based approaches.

Scene-based analysis based on generic, model-free methods has been studied from the 1980sonwards and has led to successful systems for camera pose estimation and image stabilization. Thismethod usually exploits either rigid-body scene assumptions or optic flow to estimate gross motioncharacteristics. A common approach in this area is to extract structure from motion, assuming asingle moving camera in a static scene [25, 19]. While challenges remain in these areas, commercialcodes are starting to become available to solve such problems (e.g., http://www.2d3.com/).

Most of the deep motion-related research problems in vision arise when the scene must be in-terpreted in terms of objects undergoing independent motion. Consequently, doing motion analysissuccessfully for computer vision often relies on having sufficiently expressive shape (or, more gen-erally, appearance) models to describe the objects being viewed. Certain classes of shapes havebeen successfully modeled. These include rigid, near planar shapes, and those whose contours orintensities can be modeled using linear combinations of principal components. There have also beensuccesses with 3-D jointed structures as well as 2.5-D ‘cardboard cutout’ representations. However,there remain large classes of objects which we do not yet know how to model well, including suchpragmatically important ones like people wearing loose clothing. Much of the difficulty in object

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tracking comes as much from the challenge of devising a suitable appearance model as from themotion analysis itself.

In the past most of the shape models used have been hand-crafted, parameterized models [16]and the methods developed for motion analysis have been deterministic in nature (e.g., physics-basedmethods). Recently, to overcome the obvious limitations of hand-crafted models, to generalize therepresentation of motion in terms of parameters that follow distributions, and to improve the couplingbetween motion estimation and motion recognition, statistical methods have become popular [21,11]. Such methods promise new capabilities for segmentation, parameterized shape representation,motion estimation and motion recognition.

Within this new class of statistical methods the following open research problems need to beaddressed.

Segmentation, grouping and initialization. While there has been a significant improvement inrecent years in our ability to track objects with complex appearance and motion models, the problemof initialization — the matching of an object model to this first frame of a sequence as a precursorto tracking — has lagged behind. Initialization is typically treated as a separate algorithm distinctfrom the subsequent tracking, and often relies on generic segmentation techniques. Furthermore,initialization is often attempted using a single image whereas it might be much easier to do givenseveral frames; it is clear that we need a better understanding of how to merge the problems ofinitialization and tracking to arrive at a continuum between them. Initialization is also clearly relatedto grouping methods; for example initializing a jointed-limb model can be viewed as the task ofgrouping the individually moving limbs into a coherent whole. We have as yet no good algorithmsfor incorporating this grouping into the initialization/tracking process, perhaps in order to select theform of a shape model appropriate to an unknown object. Finally, the problem of coupling shapemodels with statistical methods in an effort to automate the estimation of the priors for improvedsegmentation and model initialization has only recently begun to be studied [13].

Statistics and learning. In common with other fields such as speech recognition, the computervision community has increasingly come to understand that methods which take advantage of learnedstatistical models are more robust and powerful than those which rely on heuristics. Although thereare learning based methods for estimating low-level motion parameters, basic research needs to bedone on coupling the low level motion parameters estimated by the above methods with the semanticinterpretation of these parameters. An example of such needed integration is the problem of Gestureand Sign Language Recognition [31].

While motion analysis has so far been studied under the assumption of fairly well lit objectsthat follow the optical flow assumption, very little research has been done on motion analysis undervarying lighting conditions. This is clearly a very important problem that is essential to allow robustmotion tracking.

Multiple scales. Objects in the world are often naturally represented at multiple scales. This istrue of object shape, which may lend itself to hierarchical description; object texture: for example apiece of paper may be well represented as having uniform texture from a distance and as containing

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discrete letters close up; and object motion, which is typically well approximated by a simple linearprocess at short timescales, while these linear motions compose over longer timescales to formcomplex behaviors which may be best described using grammars over discrete spaces. We havesome algorithms for dealing with multiple scales but no good overall theory, and we do not yet fullyunderstand the conditional dependencies which relate the different scales. The study of principledstatistical algorithms to deal correctly with inference of motion over multiple scales is also in itsinfancy.

Summary. In some ways, motion analysis is one of the success stories of recent computer visionresearch. Around ten years ago memory sizes and computing speeds became adequate to deal withlengthy image sequences and since then tracking has made great strides: simple ‘blob’ trackerswhich follow the gross motion of non-overlapping objects against relatively static backgrounds arenow a standard component of robust systems which can run unattended for months. As soon as thecomplexity of the problem is increased in any dimension, however, current solutions break down,so there is plenty of research left to do, from following the motion of complex non-rigid objectsto interpreting and summarizing extended behaviors, to coupling vision-based motion trackers withother modalities, such as speech, for improved activity recognitions.

3.7 Robotics

As in the previous section, robotics must deal with motion in the ‘‘real’’ world — and not just withidealized models of such motion in a virtual world. As a consequence, besides the usual criteriasuch as efficiency and simplicity, the performance of motion algorithms in robotics must be judgedbased on how accurately they model the real world (for a class of tasks) and how robust they are tothe uncertainties.

Modeling motion. Some of the key difficulties of modeling motion in robotics include modelselection and parameter estimation. Despite extensive work in mechanics over the last few centuries,there are numerous every-day objects that robots need to reason about and manipulate that are noteasy to model using known laws. The fundamental problem lies at the fact that the parameters neededfor the models are usually unknown and difficult to acquire. For instance, our ability to model themotion of an object rolling on the ground is limited by the difficulty of knowing the surface frictionat the interface and even the inertial properties. Or, consider the problem of modeling a deformableobject, say a human organ like the liver, in surgical robotics. How is the deformation of the liver tobe modeled? In order to tackle these problems, one has to address the following issues:

• Algorithms for estimating parameters of motion models from observations. These can oftenlead to ill-posed inverse problems, requiring sophisticated numerical techniques.

• Techniques for representing, propagating, and reasoning about uncertainty in the motion.Kalman and particle filters are a couple of examples of representing and propagating theuncertainties, but we also need algorithms for incorporating these uncertainties in collisiondetection and impact problems.

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• New empirical models. For many complex problems, empirical models are the only option tomodel motion. For example, in order to model the deformation of nonuniform elastic objects,rather than reconstructing the internal structure of the object so that known models of elasticitycan be applied, it may be preferable to construct an empirical model of the deformation behaviorof the object. Essentially all friction and impact laws used in practice are empirical, and theyare applicable only for a small class of tasks.

Many of the difficulties in modeling motion are because of the the constraints that the motion mustsatisfy. The physical laws governing the motion are obvious constraints. These laws are typicallycontinuous and are modeled using differential equations. However, the motion algorithms discretizethem. Despite the recent progress in discretization of differential equations, there many issuesremain unsolved, including the ability to extend them to deformable shapes. Whether the kineticdata structures, discussed in Section 3.1, can be used to discretize differential equations remains aninteresting open question. Contact constraints are another tpe of constraints that are hard to model.For instance, simulating the roll-slide motion during smooth contact is a challenge to both collisiondetection algorithms (especially those relying on polyhedral approximations and bounding volumes)and to traditional simulation algorithms (especially those relying on constraint stabilization). Howto represent local deformations at contact is another challenging problem — localized (‘‘point’’)contacts produce very large stresses, which locally deform apparently rigid objects and can produceimportant elastic effects such as stick-slip oscillations.

Collision detection. A central problem in robotics is detecting collision between many movingobjects. It has been extensively studied both in robotics and computational geometry. The workin computational geometry has focused on theoretical approaches, which are at times complex toimplement, while the work in robotics has lacked a general approach that works well in all cases.Commonly used in practice are methods that track the closest pair of features between two movingobjects and update the closest pair by a local computation after each time step; an underlyingassumption is that, if the time step is chosen sufficiently small, then the closest pair can be updatedincrementally and efficiently. Feature trackers work well for simple convex objects, or combinationsof a small number of such. Hierarchical representations of objects have been incorporated to thisapproach to handle more complex objects, and kinetic data structures (Section 3.1) have been usedto alleviate some of the problems of the fixed-time-sampling approach. Alternatively, hierarchies ofsimple bounding volumes (spheres or boxes) have been used to enclose shapes; these hierarchies arerefined only to the coarsest level necessary to establish that the two objects do not intersect.

The above approaches, however, are not suitable for detecting collision between many movingobjects, especially if each of them is moving with different speed, because one has to performcollision testing for every pair of objects. Instead a global approach is needed. Recently a few globalapproaches are proposed for detecting collision between many moving polygons in the plane, whichmaintain a tiling of the common exterior of the polygons into flexible cells, so that the polygonsare known to be disjoint until one of the cells self-collides. They maintain additional informationthat makes it easy to predict when a cell self-collides, and to update the tiling when that occurs.Although this approach looks promising in the plane, there are many stumbling blocks in extending

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it to 3-space. Even less is known on detecting collision between deformable objects or for detectingself-intersection in a deformable object.

A different problem is decide what to do when a collision is detected. For example, in virtual-reality applications such as computer games, digital battlefields, haptics interfaces, etc., it is criticalthat reaction to collision detection is close to reality — Should an object penetrate the other? Shouldone of the objects bounce back? Should the objects deform or break? This raises several interestingquestions that combine geometry with physical models of the objects involved.

Planning motion. A typical motion-planning problem asks for computing a collision-free motionbetween two given placements of a given robot in given environment. In its simplest form, oneassumes that the environment is fully known and there are no constraints on the motion of the robotexcept that it cannot collide an obstacle. The problem is typically solved in the configuration space,in which each placement (or configuration) of the robot is mapped as a point; see [24]. The freeconfiguration space F is the subset of the configuration space at which the robot does not intersect anyobstacle. The robot can move from an initial configuration to a final configuration without intersectingan obstacle if these two configurations lie in the same connected component of F . Planning acollision-free path thus reduces to connectivity and other topological questions in F . Numerousgeneral techniques and techniques for low-dimensional configuration spaces have been developed [8,18]. This has also led to several interesting topological questions on configuration spaces, whichare typically represented as semi-algebraic sets (finite Boolean combinations of solution sets ofpolynomial equalities and inequalities).

The dimension of configuration space depends on the degrees of freedom of the robot, and it can bequite high. Computing high-dimensional configuration spaces exactly is impractical [8]. Thereforetechniques to compute an approximate representation of F are needed. Much of the difficulty inapproximating F is in understanding the topology and in simplifying the topology of F . Recently,Monte Carlo algorithms have been developed for representing F by 1-dimensional networks, calledprobabilistic road maps (PRMs) [22, 23]. Intuitively, this network is an approximate representationof the road map, a 1-dimensional network that captures the connectivity information of F . Thesemethods sample points in F and connect them by an edge if they can be connected by a simple pathinside F . Despite several heuristics to sample well the points in F , better sampling strategies areneeded to handle narrow corridors and other difficult areas in F , so that the connectivity of F ispreserved. Variants of such methods have already been described in Section 3.4. PRMs can quicklyfind a simple collision-free path between initial and final configurations in many situations. Butthere is no absolute guarantee that they will do so – in fact, very few general purpose techniques areknown for establishing that paths do not exist.

In the above discussion, we assumed that there were no constraints on the motion of the robots,which in practice is not the case. Various nonholonomic constraints such as bound on the maximumvelocity or acceleration may exist on the moving robot. Planning a collision-free in presence of theseconstraints is considerable harder, and relatively little is known except various ad-hoc approaches.Better techniques that are practical as well as theoretically sound are need for nonholonomic motionplanning.

In some applications even more challenging problems arise. If the obstacles are also moving, the

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free configuration space has to be updated dynamically. Also flexible objects such as elastic bands,rope, or cloth cannot be properly represented with a finite number of degrees of freedom. How canone represent configuration spaces of such objects is big challenge.

3.8 Spatio-Temporal Databases

Efficient indexing schemes that support various queries are central to any large database system.Most existing database systems assume that the data is constant unless it is explicitly modified. Forexample, if the value of the age field is set to 40 years, then this age is assumed to hold (i.e., 40years is returned in response to a query) until explicitly updated. Such systems are not suitablefor representing, storing, and querying continuously moving or varying objects; either the databasehas to be continuously updated or a query output will be obsolete. Motivated by applications indigital battlefields, air-traffic control, and mobile communication systems, methods for indexingdynamic attributes are needed, which guarantee efficient access to moving objects. A basic scenariois one in which the database system receives trajectories for a large population of objects capable ofcontinuous movement; new objects may arrive, existing objects may leave, and that the positions ofexisting objects may be updated dynamically. The objects, which can assume to have rigid shapes,may be represented as points. People with mobile phones, people in Internet-worked cars, airplanes,etc. The movement of the objects may be constrained by stationary or moving objects, the movementof the objects may be confined to networks (e.g., cars confined to a road network), and the motionof objects may be highly correlated. The goal is to answer various queries based on the locations,trajectories, and topology, and to explore patterns in motion of objects. Since the data is quite largeand resides on disk, one cannot scan the whole data to answer a query.

Instead of continuously updating the position of a moving object, a better approach is to representthe position as a function f (t) of time, so that changes in object position do not require any explicitchange in the database system. With this representation, the database needs to be updated only whenthe function f (t) changes. Recently there has been some work on extending the capabilities ofexisting database systems to handle moving-object databases (MOD); see, for example, [27, 28, 15].

Methods such as KDS, developed for simulating some structures/phenomena over time, are notalways suitable for answering queries on the database of moving objects. The queries might relateeither to the current configuration of objects or to a configuration in the future — in the latter case,we are asking to predict the behavior based on the current information. In either case, trackingthe current configuration of the objects would be unnecessary and expensive. Another approach tohandle motion is to regard the time as another spatial axis and reduce the problem to a static problemin one higher dimension. For example, a set of points moving in the plane can be regarded as lines inR

3. This approach has the advantage that the data structure need not be updated unless the trajectoryof an object changes. However, it suffers from a number of problems. First, the approach expectsthe trajectory of the points to be known in advance. Second, it reduces the problem to one higherdimension, which makes the algorithms slower.

Here are a few different types of queries that one may want to answer on moving objects.

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Location based queries. These queries include range and proximity queries. For example, givena rectangle R and a time t , report all objects that will be inside R at time t ; given a rectangle R anda time interval [t1, t2], report all objects that will pass through R in the time interval [t1, t2]; given apoint p and a time interval [t1, t2], report the objects that would be nearest to p in that time interval.Despite a flurry of papers on indexing moving objects for answering location-based queries, theproblem remains largely open. The theoretical results, based on partition trees and KDSs, are toocomplex to be practical. The practical indexing schemes are based on quad-trees, R-trees, and theirvariants. They do not always work well, and their performance deteriorates with time and the datasize. In order to improve the performance of an indexing scheme, Some of them have introduced thenotion of horizon that gives the duration for which the current index performs well. They suggestto rebuild the index periodically or to adjust it when the trajectories are updated (see e.g. [29]).Although these approaches provide better query performance, it is not always practical to rebuild theindex periodically.

Continuous queries. In the above queries, the query range was spatially fixed. One could alsoask queries when the location of the query range is also moving. For example, keep track of all carswithin 5 miles. Of course, such a query can be answered by asking location-based queries repeatedly,but it would be very expensive. One would like to combine KDS with procedures for location-basedapproaches to answer such queries. The known lower bounds on range-searching data structuresimply that it is hard to answer such queries efficiently in the worst case. But can one exploit theconstraints on motion to answer them efficiently?

Trajectory-based queries. These queries involve the topology of trajectories and derived infor-mation such as velocity of the objects. These queries are deemed critical, but also rather expensive.Here is a typical query: Which of two objects will be within one mile during the next ten minutes?Which of the objects are heading East, or which of the objects are moving at speed greater than65 miles/hour. An even harder query is: report all objects whose speed doubles in the next tenminutes.

Dynamization. It is an inherent characteristic that new objects arrive and that the positions of exist-ing objects are updated dynamically. Hence, the index should support fast insert, delete operations.Although basic spatial indexing schemes, such as R-trees or kd-trees, support insertion/deletion ofobjects, the problem is considerably harder for indices handling moving objects because they haveto store additional information to predict the motion of objects. This information is expensive toupdate as objects are inserted or deleted.

The current indexing schemes basically delete and re-insert a point whenever the trajectory ofan object changes. In may applications, the trajectory of an object is known in advance. Can thisinformation be used to avoid explicit deletion and re-insertion of the object.

Uncertainty. Since in many applications the position of objects is being received through a sensorsuch as GPS, uncertainty in position and velocity of the objects is inevitable. Indexing schemes thatcan handle uncertainty are needed. A simple model of uncertainty is a to allow an interval of values

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for each of the parameters defining the dynamic attributes of an object. In this model, the uncertaintyin position corresponds to a ‘ball’ in which the object can lie, and the uncertainty in velocity corre-sponds to a ‘cone’ in which the object lies. These simple models have been incorporated in a fewindexing schemes, but a more systematic study is needed. More sophisticated parametric modelshave been developed for modeling uncertainty. Although these methods have been successfully usedfor tracking moving objects, they have not been used in the context of databases. How to incorporatethem in indexing schemes is a challenging open problem.

3.9 Mobile Wireless Networks

A world of ‘anytime, anywhere, anyhow’ computing is being created by a unique combination of twopowerful technological trends: rapid component miniaturization and the emergence of high speedwireless communication. The recent boom in the use of cell phones is merely the beginning of amuch larger trend towards broadband wireless networks that carry not only voice, but all kinds ofmultimedia data. Wireless data networking presents a multitude of challenges that are distinct fromtheir fixed infrastructure networking counterpart:

• [Heterogeneity.] There are a variety of wireless technologies (e.g. modulation schemes andspectrum bands), diverse devices (e.g. laptops, PDAs, data enabled cell phones and pagers),communication protocols (e.g. several 3G standards for wireless and SMS). Finally the channelconditions under which wireless communication occurs vary widely.

• [Resource Constraints.] The nature of wireless technology and devices presents stringentresource constraints, such as the available spectrum (number of channels, codes, slots, etc.)and the power (total and received power), among others.

• [Mobility.] But the most distinguishing aspect of a wireless network is the mobility of itsclients. Dealing with mobility presents new problems (handoffs, paging clients, etc.) as wellas it complicates existing ones (such as power allocation schemes where channel conditionsvary with mobility patterns). In fact, any structuring mechanism provided by the network thatexploits the local topology of the nodes has to be adapted to motion.

Mobility and routing. A mobile ad hoc network (MANET) is an autonomous system of mobilerouters and hosts connected by wireless links. The network is self-organizing and self-configuring,with nodes establishing the necessary routes among themselves without the help of a fixed infras-tructure. Since the communicating hosts may be some distance apart, multiple network hops throughintermediate nodes may be required to complete the route. The hosts themselves act like ‘mobilerouters,’ and cooperatively forward messages not addressed to them. As these hosts move around,the routing protocol of the network must adapt its routing decisions to maintain communication. Therate of topology change may be quite dramatic in some ad hoc networks.

Since the topology of the network is constantly changing, the routing protocols must copewith frequent link failure and disconnections. The distribution of up-to-date information can easilyoverload the network, while out-of-date information can drive a network into instability. Thus, the

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shortest path routing used commonly in fixed infrastructure networks may be the wrong routingalgorithm for ad hoc networks. Instead, discovering and routing on ‘long lived’ routes may be moredesirable.

In large-scale ad hoc networks, it may be infeasible, and even undesirable, for each host tomaintain up-to-date information about the locations of all hosts and the topology of the entire network.Perhaps a more useful approach will be a statistical modeling of the movement of individual hosts. Agrand challenge in the field, which may be currently just academic, but which will have tremendousimpact on systems and performance, is the following: Can we model the mobility patterns of usersat varying levels of granularity? The question should be explored in the context of modeling arrivalpatterns of customers to service queues, web server request sizes, etc. It could be refined in manyways — e.g. does an individual users’ mobility pattern fit a small number of short random walkson the plane? Can the number of roaming users in any cell at any moment be predicted withoutunderstanding individual users’ mobility pattern? This grand challenge may involve input fromgeometers on finding a small set of attributes of the users’ trajectories on the globe that are mostpredictive of the users’ movements. One of the aspects of this grand challenge that is interesting isits ‘internet’ scale of users and resource usages.

Mobility and network layers. Another approach to study the impact of mobility on networkinginfrastructure is to study its effect on the various architectural layers of the network. We list belowfor each layer some fundamental issues that are worth exploring.

1. [Physical Layer.] How does one perform resource scheduling at the physical layer in order tooptimize performance for a given mobility pattern. For instance, how does one optimize acrossissues of power consumption, error correction, coding, rate control, channel assignment, andso on.

2. [MAC Layer.] How does one discover resources in the presence of ad-hoc mobility, such asreachability testing, topology determination, and so on.

3. [Transport Layer.] How and to what extent must one implement end-to-end TCP properties inpresence of host mobility? How does high vs. low mobility affect persistence of TCP connec-tions and fragmentation? Packet loss probabilities are significantly higher in wireless networksthan in the wired networks due to widely varying channel conditions that are compounded byusers’ mobility.

Specifically, the TCP reacts quite violently to packet losses, with its exponential backoff mech-anism, because it interprets the loss of packet to mean congestion in the network. However,in the wireless networks, the packet losses are just as likely the result of transmission loss ascongestion, so how should the TCP rate control mechanism be modified?

4. [Network Layer.] How should one balance the cost of ‘redirection’ vs. ‘paging,’ in networksfor accessing a user by better understanding of mobility models?

5. [Application Layer.] How does one support location specific queries on databases of yellowpages or services? How does one support a geographically connected community of users

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or ‘buddies’? These involve testing geographic neighborhoods based on distance constraintswhere distance updates are obtained by pinging.

Many additional issues are likely to arise as wireless data networking and web evolve to becomemore widespread. While many of the issues listed above are algorithmic, they are best exploredby collaboration with electrical engineers and computer scientists who work on different layers ofwireless networking in order to be effective. Such collaborations represent an opportunity for thetheoretical computer science community to formulate and solve the fundamental problems of interestin mobility and related issues in wireless networking. Finally, there is clearly a large overlap betweenthe problems discussed here and those in Section 3.8 on spatio-temporal queries for mobile data.

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4 Challenges

In this section we review a number of challenging issues that cut across many areas.

Motion representation and analysis. This covers most of the topics we have already presentedin earlier sections:

Uncertainty. Whenever we try to model motion in the physical world we must deal with uncertainty.So far only the computer vision and robotics communities have developed refined tools forthe probabilistic treatment of object motion. Such ideas could also prove very very useful inthe temporal database and networking domains.

Robustness. Many geometric algorithms suffer from robustness problems because of mismatchbetween fast floating-point computer arithmetic, which is finite precision, and the semanticsof real analysis. Algorithms fail when an approximate computation yields an incorrect oreven inconsistent qualitative property (often a topological property). In the context of kineticalgorithms, uncertainty in the input makes the problem even more acute.

Aggregation. Related to uncertainty is the issue of statistical models for representing aggregatemotion, say the motion of cars on the freeway, or a flock of birds. We may be able to reasonreliably about the high-level behavior of the system, even though we have large uncertaintyabout the motion of each individual object.

Simplification. While shape simplification methods are highly developed, much less is known aboutmotion simplification. A molecular dynamics simulation, for example, includes all the atomicvibrations caused by thermal noise, which can obscure the large-scale structure of the motionthat we want to understand.

Hierarchical representations of complex motion. Related to simplification is the issue of devisinghierarchical and multi-resolution motion representations.

Marrying the continuous and the discrete. Most physical systems evolve following continuousevolution laws, yet their evolution is punctuated by discrete events, such as collisions, that canalter these laws. Historically the communities that have studied the continuous and discreteaspects of this problem have been distinct (scientific computing vs. discrete algorithms). Amuch tighter integration between the two can lead to substantial progress.

Developing efficient algorithms. Clearly a lot remains to be accomplished on developing efficientmotion algorithms and the tools for their analysis.

Trade-off between realism and efficiency. Little has been done so far to formally explore trade-offs between accuracy and efficiency. Such analyses can benefit all of the areas we have beendiscussing.

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Decentralization. Distributed motion algorithms are equally important for large-scale simulationsand for low-power mobile devices in situations such as ad hoc networking or sensor nets.

Querying motion. Many fundamental questions remain on how best to sense and organize mobiledata when the goal is to answer certain queries about the mobile system (and there is no needto know the full state of the system in between).

Motion Integration. Though each of the areas we discussed has good reasons for the motionrepresentations it has chosen, this diversity hinders the development of end-to-end systems wheretechniques form several disciplines need to process motion data in an integrated fashion. Currentlybig gaps exist between the motion representations best suited for visual motion analysis and thosefor motion generation, for example. We feel it is important to define a number of fundamentalmotion representations that can be used across areas and which can form the basis for implementingintegrated applications.

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5 Recommendations

• Funding opportunities are needed to encourage further work in these areas, either as a separateinitiative or as continued funding from the relevant areas within NSF and DOD.

• To facilitate cross-disciplinary research, we recommend a solicitation on the theme of anintegrated, end-to-end system dealing with motion. The focus of the research will be thecollaboration of motion researchers from different communities in a system that exploitsmotion representations and algorithms across several fields. Example might be: real-timerobotic mimesis based on motion capture (a robotic actor imitating the motions of a realactor in real-time), real-time monitoring of ad hoc mobile or cellular network performance,multiple vehicle tracking using a distributed sensor net, the simulation of deformable objectsin large-area contact, multi-resolution molecular dynamics, etc.

• It seems premature to establish a journal or annual conference series in this area, but at thevery least there should be workshops on modeling motion at regular time intervals, say, onceevery other year. The last workshop was an invitation-only, direction-finding session; what isneeded now is a forum for collecting new work in the area, unifying the work being done indifferent communities, and fostering interdisciplinary collaboration.

Agarwal and Guibas are planning to organize a week-long workshop in Fall 2002 at DIMACS,Rutgers University, New Jersey, as part of the special year on computational geometry. Itwill be an open workshop in which participants will present recent work on motion modeling.There will also be plenty of time for participants to discuss open problems and to work together.

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